A knowledge of the design principles of neuronal processes is fundamental to understanding the functional interactions that exist between neurons. The proposed study is to develop two unified mathematical models of dendritic growth called the tree-pattern and tree growth branching process models. The quantitative features which will be implemented in these models include the geometric branching pattern of dendritic trees, the lengths and the diameters of the branches, and the time development of these features. To estimate parameters of the mathematical models and to test predictions of this parameterized model against observations of real dendritic trees, a database of dendritic trees will be collected from developing rats, adults rats, and adult rhesus monkeys. Dendritic data will be collected from the hippocampus and dentate gyrus in vitro and in vivo. For in vivo studies of dendritic growth in the rats, pyramidal neurons of the hippocampus and granule cells of the dentate gyrus will be digitized at postnatal days 5, 10, 15, 24, 48, and 90. For in vitro studies, the dendrites of hippocampal neurons will be measured at postplating days 2, 4, 6, 8, and 12. A mathematical model reduces the complexity of a problem to a few model parameters. Once a model has been established, only the parameters have to be estimated for each new experimental situation. Thus, these mathematical models can reduce the number of biological experiments. The proposed studies should enhance the understanding of dendritic growth behavior, help facilitate the design of computational neuronal models, and develop mathematical and statistical methodology for reliable comparisons of dendritic growth in different environments.